Cosmology Omega Lambda Derivation Rs Consistent With Planck

The Planck consistency claim

In cosmology, the dark energy fraction Ω_Λ is the portion of the universe's energy density that drives its accelerated expansion. The Planck 2018 mission measured it as 0.6889 with an uncertainty of 0.0056. The Recognition Science framework's library of formal theorems contains a result, rs_consistent_with_planck, which proves that its own derived value for this fraction lies within 0.0112 of the measured one, a span of two standard deviations.

The framework's value comes from a specific formula. It starts with a raw fraction of 11/16, or 0.6875, which it derives from a mode budget: 11 saturated modes out of 16 possible from a four-bit addressing of an eight-tick cycle. It then subtracts a correction of α/π, where α is the measured electromagnetic coupling constant, to account for modes that are electromagnetically active. This yields a final value of approximately 0.6852, which falls inside the interval (0.683, 0.686) that the theorem certifies. The consistency theorem states that this interval is within the two-sigma window of the Planck 2018 result.

The derivation uses exactly one measured input: the CODATA value of the fine-structure constant α. The 11/16 fraction and the shape of the correction are framework-internal results. An earlier version of the derivation replaced the measured α with a constructed value to claim zero free parameters, but that edit was reverted because the constructed value is an identification, not a derived coupling, and its first-order value is excluded by measurement by more than 30,000 standard deviations. The current form is honest about its single empirical input.

What the theorem does not claim is just as important. It does not prove that the framework's value is the true cosmological constant fraction; it proves only consistency with the Planck measurement within two sigma. It does not derive the fine-structure constant from first principles; that remains an open problem. The theorem is a check that the framework's internal structure produces a number in the right ballpark, not a derivation of the universe's expansion history.

/-- The RS value is consistent with Planck 2018 at the 2σ level.
    The computed interval (0.683, 0.686) is within 1σ of 0.6889 ± 0.0056 = (0.6833, 0.6945).
    Scope honesty (2026-07-29): `omega_lambda` carries the MEASURED α as an input
    (`em_correction = α/π`, α being an irreducible boundary datum per
    `AlphaGenesis.KappaGammaIrreducibility`), so this is a consistency check of
    the static term `11/16`, not a parameter-free prediction. The word
    "prediction" returns when α is derived. -/
theorem rs_consistent_with_planck :
    |omega_lambda - omega_lambda_planck2018| < 2 * omega_lambda_planck_err := by
  unfold omega_lambda_planck2018 omega_lambda_planck_err
  have h_twosigma : 2 * (0.0056 : ℝ) = 0.0112 := by norm_num
  rw [h_twosigma, abs_lt]
  refine ⟨?_, ?_⟩
  · have := omega_lambda_gt_683
    linarith
  · have := omega_lambda_lt_686
    linarith

/-- Canonical form: Ω_Λ = 11/16 − α/π with the measured CODATA α. -/
theorem omega_lambda_canonical_form :
    omega_lambda = 11/16 - Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
  rw [omega_lambda_one_measured_input]
  unfold N_modes_saturated tick_addressing
  norm_num

/-- Ω_Λ decomposes as integer combinatorics minus the measured-α correction:
    one measured input, everything else structural. -/
theorem omega_lambda_one_measured_input :
    omega_lambda = (N_modes_saturated : ℝ) / (tick_addressing : ℝ)
      - Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
  unfold omega_lambda omega_raw em_correction
  rfl

/-- Ω_Λ ∈ (0.683, 0.686). -/
theorem omega_lambda_interval : 0.683 < omega_lambda ∧ omega_lambda < 0.686 :=
  ⟨omega_lambda_gt_683, omega_lambda_lt_686⟩

What this page does not claim

The theorem does not prove the framework's value is the true cosmological constant fraction. The theorem does not derive the fine-structure constant from first principles. The theorem does not claim the framework's value is more precise than the Planck measurement.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/OmegaLambdaDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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